Answer
Solution set: $[0,3]\cup[5,\infty)$
Work Step by Step
Follow the "Procedure for Solving Polynomial lnequalities",\ p.412:
1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$
where $f$ is a polynomial function.
$x(3-x)(x-5) \leq 0$
$f(x)=x(3-x)(x-5)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$x(3-x)(x-5)=0$
$x=0$ or $x=3$ or $x=5$
3. Locate these boundary points on a number line, thereby dividing the number line into intervals.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \leq 0 ? \\
& & a(3-a)(a-5) & \\
(-\infty,0) & -1 & (-)(+)(-) & F\\
(0,3) & 1 & (+)(+)(-) & T\\
(3,5) & 4 & (+)(-)(+) & F\\
(5,\infty) & 10 & (+)(-)(+) & T
\end{array}$
5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points.
Solution set: $[0,3]\cup[5,\infty)$
